ZSTEMR computes selected eigenvalues and, optionally, eigenvectors


 of a real symmetric tridiagonal matrix T. Any such unreduced matrix has


 a well defined set of pairwise different real eigenvalues, the corresponding


 real eigenvectors are pairwise orthogonal.


 The spectrum may be computed either completely or partially by specifying


 either an interval (VL,VU] or a range of indices IL:IU for the desired


 eigenvalues.


 Depending on the number of desired eigenvalues, these are computed either


 by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are


 computed by the use of various suitable L D L^T factorizations near clusters


 of close eigenvalues (referred to as RRRs, Relatively Robust


 Representations). An informal sketch of the algorithm follows.


 For each unreduced block (submatrix) of T,


    (a) Compute T - sigma I  = L D L^T, so that L and D


        define all the wanted eigenvalues to high relative accuracy.


        This means that small relative changes in the entries of D and L


        cause only small relative changes in the eigenvalues and


        eigenvectors. The standard (unfactored) representation of the


        tridiagonal matrix T does not have this property in general.


    (b) Compute the eigenvalues to suitable accuracy.


        If the eigenvectors are desired, the algorithm attains full


        accuracy of the computed eigenvalues only right before


        the corresponding vectors have to be computed, see steps c) and d).


    (c) For each cluster of close eigenvalues, select a new


        shift close to the cluster, find a new factorization, and refine


        the shifted eigenvalues to suitable accuracy.


    (d) For each eigenvalue with a large enough relative separation compute


        the corresponding eigenvector by forming a rank revealing twisted


        factorization. Go back to (c) for any clusters that remain.


 For more details, see:


 - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations


   to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"


   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.


 - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and


   Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,


   2004.  Also LAPACK Working Note 154.


 - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric


   tridiagonal eigenvalue/eigenvector problem",


   Computer Science Division Technical Report No. UCB/CSD-97-971,


   UC Berkeley, May 1997.


 Further Details


 1.ZSTEMR works only on machines which follow IEEE-754


 floating-point standard in their handling of infinities and NaNs.


 This permits the use of efficient inner loops avoiding a check for


 zero divisors.


 2. LAPACK routines can be used to reduce a complex Hermitean matrix to


 real symmetric tridiagonal form.


 (Any complex Hermitean tridiagonal matrix has real values on its diagonal


 and potentially complex numbers on its off-diagonals. By applying a


 similarity transform with an appropriate diagonal matrix


 diag(1,e^{i hy_1},


... , e^{i hy_{n-1}}),


the complex Hermitean


 matrix can be transformed into a real symmetric matrix and complex


 arithmetic can be entirely avoided.)


 While the eigenvectors of the real symmetric tridiagonal matrix are real,


 the eigenvectors of original complex Hermitean matrix have complex entries


 in general.


 Since LAPACK drivers overwrite the matrix data with the eigenvectors,


 ZSTEMR accepts complex workspace to facilitate interoperability


 with ZUNMTR or ZUPMTR.

JOBZ

          JOBZ is CHARACTER*1


          = 'N':  Compute eigenvalues only;


          = 'V':  Compute eigenvalues and eigenvectors.

RANGE

          RANGE is CHARACTER*1


          = 'A': all eigenvalues will be found.


          = 'V': all eigenvalues in the half-open interval (VL,VU]


                 will be found.


          = 'I': the IL-th through IU-th eigenvalues will be found.

N

          N is INTEGER


          The order of the matrix.  N >= 0.

D

          D is DOUBLE PRECISION array, dimension (N)


          On entry, the N diagonal elements of the tridiagonal matrix


          T. On exit, D is overwritten.

E

          E is DOUBLE PRECISION array, dimension (N)


          On entry, the (N-1) subdiagonal elements of the tridiagonal


          matrix T in elements 1 to N-1 of E. E(N) need not be set on


          input, but is used internally as workspace.


          On exit, E is overwritten.

VL

          VL is DOUBLE PRECISION


          If RANGE='V', the lower bound of the interval to


          be searched for eigenvalues. VL < VU.


          Not referenced if RANGE = 'A' or 'I'.

VU

          VU is DOUBLE PRECISION


          If RANGE='V', the upper bound of the interval to


          be searched for eigenvalues. VL < VU.


          Not referenced if RANGE = 'A' or 'I'.

IL

          IL is INTEGER


          If RANGE='I', the index of the


          smallest eigenvalue to be returned.


          1 <= IL <= IU <= N, if N > 0.


          Not referenced if RANGE = 'A' or 'V'.

IU

          IU is INTEGER


          If RANGE='I', the index of the


          largest eigenvalue to be returned.


          1 <= IL <= IU <= N, if N > 0.


          Not referenced if RANGE = 'A' or 'V'.

M

          M is INTEGER


          The total number of eigenvalues found.  0 <= M <= N.


          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W

          W is DOUBLE PRECISION array, dimension (N)


          The first M elements contain the selected eigenvalues in


          ascending order.

Z

          Z is COMPLEX*16 array, dimension (LDZ, max(1,M) )


          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z


          contain the orthonormal eigenvectors of the matrix T


          corresponding to the selected eigenvalues, with the i-th


          column of Z holding the eigenvector associated with W(i).


          If JOBZ = 'N', then Z is not referenced.


          Note: the user must ensure that at least max(1,M) columns are


          supplied in the array Z; if RANGE = 'V', the exact value of M


          is not known in advance and can be computed with a workspace


          query by setting NZC = -1, see below.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          JOBZ = 'V', then LDZ >= max(1,N).

NZC

          NZC is INTEGER


          The number of eigenvectors to be held in the array Z.


          If RANGE = 'A', then NZC >= max(1,N).


          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].


          If RANGE = 'I', then NZC >= IU-IL+1.


          If NZC = -1, then a workspace query is assumed; the


          routine calculates the number of columns of the array Z that


          are needed to hold the eigenvectors.


          This value is returned as the first entry of the Z array, and


          no error message related to NZC is issued by XERBLA.

ISUPPZ

          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )


          The support of the eigenvectors in Z, i.e., the indices


          indicating the nonzero elements in Z. The i-th computed eigenvector


          is nonzero only in elements ISUPPZ( 2*i-1 ) through


          ISUPPZ( 2*i ). This is relevant in the case when the matrix


          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

TRYRAC

          TRYRAC is LOGICAL


          If TRYRAC.EQ..TRUE., indicates that the code should check whether


          the tridiagonal matrix defines its eigenvalues to high relative


          accuracy.  If so, the code uses relative-accuracy preserving


          algorithms that might be (a bit) slower depending on the matrix.


          If the matrix does not define its eigenvalues to high relative


          accuracy, the code can uses possibly faster algorithms.


          If TRYRAC.EQ..FALSE., the code is not required to guarantee


          relatively accurate eigenvalues and can use the fastest possible


          techniques.


          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix


          does not define its eigenvalues to high relative accuracy.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)


          On exit, if INFO = 0, WORK(1) returns the optimal


          (and minimal) LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK. LWORK >= max(1,18*N)


          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (LIWORK)


          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER


          The dimension of the array IWORK.  LIWORK >= max(1,10*N)


          if the eigenvectors are desired, and LIWORK >= max(1,8*N)


          if only the eigenvalues are to be computed.


          If LIWORK = -1, then a workspace query is assumed; the


          routine only calculates the optimal size of the IWORK array,


          returns this value as the first entry of the IWORK array, and


          no error message related to LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          On exit, INFO


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value


          > 0:  if INFO = 1X, internal error in DLARRE,


                if INFO = 2X, internal error in ZLARRV.


                Here, the digit X = ABS( IINFO ) < 10, where IINFO is


                the nonzero error code returned by DLARRE or


                ZLARRV, respectively.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA